Properties

Label 2-690-5.2-c2-0-1
Degree $2$
Conductor $690$
Sign $-0.891 + 0.452i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (−3.81 + 3.22i)5-s + 2.44·6-s + (3.99 + 3.99i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−7.04 − 0.590i)10-s − 16.7·11-s + (2.44 + 2.44i)12-s + (−7.41 + 7.41i)13-s + 7.98i·14-s + (−0.723 + 8.62i)15-s − 4·16-s + (−17.8 − 17.8i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.763 + 0.645i)5-s + 0.408·6-s + (0.570 + 0.570i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.704 − 0.0590i)10-s − 1.51·11-s + (0.204 + 0.204i)12-s + (−0.570 + 0.570i)13-s + 0.570i·14-s + (−0.0482 + 0.575i)15-s − 0.250·16-s + (−1.05 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.891 + 0.452i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4291307007\)
\(L(\frac12)\) \(\approx\) \(0.4291307007\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (3.81 - 3.22i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (-3.99 - 3.99i)T + 49iT^{2} \)
11 \( 1 + 16.7T + 121T^{2} \)
13 \( 1 + (7.41 - 7.41i)T - 169iT^{2} \)
17 \( 1 + (17.8 + 17.8i)T + 289iT^{2} \)
19 \( 1 + 18.0iT - 361T^{2} \)
29 \( 1 - 17.7iT - 841T^{2} \)
31 \( 1 - 2.15T + 961T^{2} \)
37 \( 1 + (-34.1 - 34.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 55.9T + 1.68e3T^{2} \)
43 \( 1 + (25.9 - 25.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (1.36 + 1.36i)T + 2.20e3iT^{2} \)
53 \( 1 + (17.0 - 17.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 72.1iT - 3.48e3T^{2} \)
61 \( 1 + 67.4T + 3.72e3T^{2} \)
67 \( 1 + (-29.6 - 29.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 21.4T + 5.04e3T^{2} \)
73 \( 1 + (73.0 - 73.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 86.1iT - 6.24e3T^{2} \)
83 \( 1 + (105. - 105. i)T - 6.88e3iT^{2} \)
89 \( 1 - 86.6iT - 7.92e3T^{2} \)
97 \( 1 + (42.2 + 42.2i)T + 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.11900064495871754808607043056, −9.817064372328351220271411231162, −8.660242915646263198946324645845, −8.050022550130990500450882571713, −7.17272118092165870733775222075, −6.63958482690766659405187127157, −5.15883082414138207838648573126, −4.55794970741028643591788689547, −3.00174734812399550577118371183, −2.38875379639431380932255339781, 0.11226075244447698180644956035, 1.86092673422857230281718239594, 3.17310528410218447389197787654, 4.23393185757953166940005740224, 4.83951499157911674048413457410, 5.81345676193364746209092794036, 7.47063514369303918162857867167, 8.024324813953008033106038707942, 8.839237261639950774290316762014, 10.10979490877962465488733754160

Graph of the $Z$-function along the critical line